1914]
SPHERES OF SMALL RELATIVE INDEX
225
For values of n much greater, (22) is sufficiently represented by nV2/16, or m"; simply. It appears that there is no tendency to a falling-off in the scattering, such as would allow an increased transmission.
In order to make sure that the special choice of values for <m has not masked a periodicity, I have calculated also the results when n is even. Here sin 2m = 0 and cos 2m = + ], so that (21) reduces to
112 (1 or 0)
n~7r~
WIT"
RA.
° _ 4) [7 + log (nvr/2) _ Ci (rwr/2)]. (23)
The following are required:
n Ci (ttTr/2) n Ci (717T/2)
2 4 6 + 0-0738 -0-0224 + 0-0106 - 8 10 -0-0061 +0-0040
of which the first is obtained by interpolation from Glaisher's Table VI, and the remainder directly from (19). Thus:
1
n. (28) ! n (23)
2 0-7007 8 32-336
4 i 6-1077 10 53-477
6 i 16-156
The better to exhibit the course of the calculation, the actual values of the several terms of (23) when n = 10 may be given. We have
1 12 fi27T2
_ -^ = _ 0-11348, -v£- = 61-685,
n 7T~ 10
, 64 ,
7 + log (7T/2) + log n - Ci (7i7r/2) = 0'57722 + 0-45158 + 2'30259 - 0'0040
= 13-094,
so that
4 —
{7 + log (n7r/2) - Ci (n7r/2)} = 13'094.
It will be seen that from this onwards the term nV/16, viz., m2, greatly preponderates; and this is the term which leads to the limiting form (20). ,
The values of 2JS/X concerned in the above are very moderate. Thus, n = 10, making m = 4,7rR/\ = IQ-irfi, gives 2R/\ = 5/4 only. Neither below
E. VI. 15 - du ............................ (17)